Calculating effective mass when Spring itself is massive in Mechanics problems.

So, this is a post regarding a general result which is of a lot of help if remembered in Mechanics problems. The derivation is of how to calculative effective mass of block and spring attached if spring is massive.

Let's consider the mass of spring to be $m$ .

So let's begin,

So let us take a small element of mass $dm$ of the spring, so total energy can be calculated by adding all the length elements' kinetic energy, and requires the following integral:

$E = \int \dfrac{1}{2}u^{2} dm$

Since spring is uniform,

$dm = (\dfrac{dy}{L})m$

$E = \int_{0}^{L} \dfrac{1}{2}u^{2}(\dfrac{dy}{L})m$

The velocity of each mass element of the spring is directly proportional to its length, i.e.

$u = (\dfrac {vy}{L})$

$E = \dfrac{1}{2} \dfrac{m}{L} \int_{0}^{L} (\dfrac{vy}{L})^{2} dy$

$E= \dfrac{1}{2} \dfrac{m}{3} v^{2}$

Comparing to the expected original kinetic energy formula,

$K.E. = \dfrac{1}{2}mv^{2}$

we can conclude that effective mass of spring in this case is $\dfrac{m}{3}$

The EFFECTIVE MASS of the spring in a spring-mass system when using an ideal spring of uniform linear density is $\dfrac{1}{3}$ of the mass of the spring.

And we're done.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Calculating effective mass when Spring itself is massive in Mechanics problems.

Comments